Global weak solutions to the generalized Proudman-Johnson equation
Chien-Hong Cho Marcus Wunsch
We consider the generalized Proudman-Johnson equation, in which an artificial parameter $a$ controlling the impact of convection was introduced to the Proudman-Johnson equation ([33]). In the present paper, we are going to show that there are global weak solutions to the generalized Proudman-Johnson equation for certain parameter $a$'s.
keywords: global weak solutions. The generalized Proudman-Johnson equation
On the Hunter--Saxton system
Marcus Wunsch
We show local existence of solutions to a two-component Hunter--Saxton system. Moreover, we prove that the slopes of solutions can become unbounded. Finally, if initial data satisfy appropriate smallness conditions, the associated flow is global.
keywords: Two-component Hunter--Saxton system global existence in time. local well-posedness blow-up phenomena
Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation
Hisashi Okamoto Takashi Sakajo Marcus Wunsch
Steady-states and traveling-waves of the generalized Constantin--Lax--Majda equation are computed and their asymptotic behavior is described. Their relation with possible blow-up and the Benjamin--Ono equation is discussed.
keywords: generalized Constantin--Lax--Majda equation. Steady-state traveling wave
The geometry of a vorticity model equation
Joachim Escher Boris Kolev Marcus Wunsch
We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
keywords: Euler equation on diffeomorphisms group of the circle. CLM equation

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